"I'm thinking of an animal...": Public Announcements and Gricean Implicatures

Yesterday walking home from school, G. and I mutually stumbled upon a new game. Neither of us had ever played it before (or even heard of it before), and yet it was one of those games where it was completely obvious what the right rules were. The game is "I'm thinking of an animal", and it goes something like this:

1. Person 1: "I'm thinking of an animal that swims."
2. Person 2: "A fish."
3. Person 1: "I'm thinking of an animal that swims and flies."
4. Person 2: "A flying fish."
5. Person 1: "I'm thinking of an animal that swims and flies and bends its head underwater to eat."
6. Person 2: "A swan?"
7. Person 1: "I'm thinking of an animal that swims, flies, bends its head underwater to eat, and has many colors."
8. Person 2: "A duck!"
9. Person 1: "Yaaay!!!"

Or another version:

1. Person 1: "I'm thinking of an animal that lives in the desert."
2. Person 2: "A camel!"
3. Person 1: "Oh, I thought that one would be harder."

Or this fun instance:

1. Person 1: "I'm thinking of an animal that loves to fly."
2. Person 2: "A bird."
3. Person 1: "I'm thinking of an animal that loves to fly and has clear wings."
4. Person 2: "A fly?"
5. Person 1: "I'm thinking of an animal that loves to fly, has clear wings, and isn't true."
6. Person 2: "Oh, a fairy!"
7. Person 1: "Yaaaayyyy!!!"

(It's a great game).

Of course, #occupationalhazard, I started thinking about what sort of strategies are being used to play such a game, and what sort of properties make certain moves good or bad, and how one would go about modeling this. Clearly, Player 1 is making public announcements that successively carve pieces out of Player 2's epistemic space until Player 2 is left with either the right option (in which case Player 2 wins) or no option (in which case Player 2 loses). But winning and losing isn't merely a matter of guessing the right answer or not; if Player 2 gets the answer right after the first clue, then it's a victory but not a very satisfying one; Player 1 should've been more strategic.

This strategic aspect of Player 1's choice of clues to give is intimately tied up with Grice's conversational maxims, specifically the maxim of quantity:

The maxim of quantity: Be as informative as one possibly can, giving as much information as is needed, and no more.

If Player 1 played according to this maxim, she'd offer as a first clue something like "I'm thinking of an animal that moos" or "I'm think of an animal that is called 'cow' in English" -- and these entirely defeat the purpose of the game. The first clue does need to narrow down the possibility space somewhat ("I'm thinking of an animal" is of no use, nor even is "I'm thinking of an animal that is alive" -- though the complement of that, "I'm thinking of an animal that is extinct", is a good first clue!), but after that, the best way to play the game is for Player 1 with each successive clue to:

• Make an announcement that clearly excludes the previous wrong guess of Player 2
• Exclude as little else as possible.

Bonus points if you can give clues that misdirect, e.g. "lives in England" followed by "has two legs" and a few others before following up with "talks", at which point I finally realised the answer wasn't any kind of bird, but rather humans! Or when I did "goes very slow", "likes water", "has a shell", hoping to trick her into saying snail when I was aiming for tortoise.

Another dimension that makes the game interesting is the presence of common knowledge amongst the group of players that is unlikely to be shared by people outside that group. For instance "you cuddle with this animal at night" and "I have a pair of socks with them on" are unlikely to elicit "lemur" from many other pairs of players, whereas it was a dead give away for us.

So, what then is the best strategy? The first public announcement Player 1 makes needs to be understood to be carving away a large portion of the candidates, while still leaving a large enough set behind, and successive announcements should be made flouting the maxim of quantity as much as possible, that is, one should try to say as little as possible with every given announcement, all the while being sensitive to the group knowledge of the players.

It's a fun game (and easily extendible to "I'm thinking of an X" for pretty much any X), and because Player 1 (who knows the answer) controls the rate of reduction of the model, it is less open ended than similar games like "20 Questions", where it is Player 2 (who doesn't know the answer but asks the questions) who controls the rate of reduction. You should play it!

Edit: Just after I published this, Rineke Verbrugge over on FB mentioned an excellent paper touching on many of the aspects illustrated in "I'm thinking of an animal...": Rineke Verbrugge & Lisette Mol, "Learning to Apply Theory of Mind", Journal of Logic, Language and Information 17 (2008): 489-511.